What is the argument of the complex number z = -1 - i?

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Q: What is the argument of the complex number z = -1 - i?

Solution: The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = -3π/4.

Steps: 8

Step 1: Identify the complex number z = -1 - i. This means the real part is -1 and the imaginary part is -1.
Step 2: Use the formula for the argument θ of a complex number, which is θ = tan^(-1)(imaginary part / real part).
Step 3: Substitute the values: θ = tan^(-1)(-1 / -1).
Step 4: Simplify the fraction: -1 / -1 = 1, so now we have θ = tan^(-1)(1).
Step 5: Find the angle whose tangent is 1. This angle is π/4 radians.
Step 6: Determine the correct quadrant for the angle. Since both the real and imaginary parts are negative, the complex number is in the third quadrant.
Step 7: In the third quadrant, the angle is π/4 plus π (to move to the third quadrant), which gives us θ = π/4 + π = 5π/4.
Step 8: Alternatively, we can express this angle as -3π/4, since angles can be represented in multiple ways.